Similarity Solutions of a Non-linear Diffusion Equation

In several physical contexts the equations for the dispersionof a buoyant contaminant can be approximated by the Erdogan-Chatwin(1967) equation {dot}c = {dot}_y{[D_o + ({dot}_yc)²D₂]{dot}_yc}. Here it is shown that in the limit of strong non-linearity (i.e.D_o = 0) there are similarity solutions for a concentration jumpand for a finite discharge. A stability analysis for the latterproblem involves a new family of orthogonal polynomials Y_n(z)where (1 – z⁴)Y – 6z³Y + n(n + 5)z² Y_n = 0 and the degree n is restricted to the values 0, 1, 4, 5, 8,9,.... A numerical solution of the Erdogan-Chatwin equationis given which describes the transition between the non-linearand linear (Gaussian) similarity solutions.     

Gluing torsion endo-permutation modules

Let k be a field of characteristic p, and let P be a finitep-group, where p is an odd prime. In this paper, we considerthe problem of gluing compatible families of endo-permutationmodules: being given a torsion element M_Q in the Dade groupD(N_P(Q)/Q), for each non-trivial subgroup Q of P, subject toobvious compatibility conditions, we show that it is alwayspossible to find an element M in the Dade group of P such that for all Q, but that Mneed not be a torsion element of D(P). The obstruction to thisis controlled by an element in the zeroth cohomology group over₂ of the poset of elementary abelian subgroups of P of rankat least 2. We also give an example of a similar situation,when M_Q is only given for centric subgroups Q of P. Moreover,general results about biset functors and the Dade functor aregiven in two appendices.     

Extended Generalized Hexagons and the Suzuki Chain

Four extended generalized hexagons related to the simple groupsG₂(2)', PSU₄(3), HJ and Suz are characterized by the conditionthat any triple of points {x, y, z} is a clique of the pointgraph not in a circle of the extended hexagon if and only ifthe distance of y and z in the residue at x is 3.     

Error Control with Polynomial Approximations

A detailed analysis is given of the accumulation of errors whichmay occur in evaluating a polynomial approximation to a givenfunction. Both backward recursion using untransformed Chebyshevexpansions and the much faster nested multiplication using thetransformed simple polynomial form are treated. Two types ofarithmetic are dealt with covering most current machines. Forthe case of polynomials with coefficients of the same sign orstrictly alternating signs, a situation which is of considerablepractical importance in polynomial approximation of mathematicalfunctions, we show that controlling relative error requiresthe ratios |y|_max/|y|_min and |y'|_max/|y|_min to be kept small.Experimental verification of these effects is given based onexpansions available in the literature or produced by the authorsfor the Bessel function I₀(x).     

Fast Solution of Vandermonde-Like Systems Involving Orthogonal Polynomials

Consider the (n + 1) ? (n + 1) Vandermonde-like matrix P=[p_i-1(_j-1)],where the polynomials p_o(x), ..., p_n(x) satisfy a three-termrecurrence relation. We develop algorithms for solving the primaland dual systems, Px = b and P^Ta = f respectively, in O(n²)arithmetic operations and O(n) elements of storage. These algorithmsgeneralize those of Bj?rck & Pereyra which apply to themonomial case p_i(x). When the p_i(x) are the Chebyshev polynomials,the algorithms are shown to be numerically unstable. However,it is found empirically that the addition of just one step ofiterative refinement is, in single precision, enough to makethe algorithms numerically stable.     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

High-accuracy P-stable Methods for y' = f(t, y)

We obtain a one-parameter family of sixth-order P-stable methodsfor the numerical integration of periodic or near-periodic differentialequations that are defined by initial-value problems of theform: y^" = f(t, y), y(t₀)= y₀, y^'(t₀)= y^’₀. Our P-stablemethods are symmetric and involve three function evaluationsper step (periteration, in case f(t, y) is non-linear in y).For non-linear problems, starting values for the solution ofthe implicit equations by modified Newton's method are suggestedand illustrated by an example.     

A Note on the WKB Method for Difference Equations

We consider the asymptotic solution of the second-order differenceequation y_n + 1 –2y_n + y_n–1 + Q_ny_n = 0, where Q_n= N^–Q(n/N), 0 < < 2, Q(s) being a differentiablefunction of s, and N a large parameter such that Q(n/N) variesby order unity as n varies by order N. A discrete WKB methodis proposed, the form of the asymptotic expansion being similarto that used in the conventional WKB method. A particular Q(s)is studied, for which results of the discrete WKB method arein agreement with the results from the approach due to Bremmer(1951).     

A division theorem for real analytic functions

We characterize those homogeneous polynomials P [z₁, ... ,z_d] for which the principal ideal (P) = P · A(^d) is complementedin A(^d) or, equivalently, those which admit a continuous lineardivision operator. The condition is the same as that which characterizes,among the homogeneous polynomials, those which are nonellipticand for which P(D) is surjective in A(^d), and those for whichP(D) admits a continuous linear right inverse in C(^d). It dependsonly on the type of real singularities.     

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphicpartial differential operators with the principal part Q_2p(D)essentially being the (complex) Laplace operator to a power,^p. We provide inital data on a singular conic divisor givenby P = 0, where P is a homogeneous polynomial of degree 2p.We show that this problem is uniquely solvable if the polynomialP is elliptic, in a certain sense, with respect to the principalpart Q_2p(D).     

The density of prime divisors in the arithmetic dynamics of quadratic polynomials

Let f [x], and consider the recurrence given by a_n = f(a_{n –1}), with a₀ . Denote by P(f, a₀) the set of prime divisorsof this recurrence, that is, the set of primes dividing at leastone non-zero term, and denote the natural density of this setby D(P(f, a₀)). The problem of determining D(P(f, a₀)) whenf is linear has attracted significant study, although it remainsunresolved in full generality. In this paper, we consider thecase of f quadratic, where previously D(P(f, a₀)) was knownonly in a few cases. We show that D(P(f, a₀)) = 0 regardlessof a₀ for four infinite families of f, including f = x² + k,k \{–1}. The proof relies on tools from group theoryand probability theory to formulate a sufficient condition forD(P(f, a₀)) = 0 in terms of arithmetic properties of the forwardorbit of the critical point of f. This provides an analogy toresults in real and complex dynamics, where analytic propertiesof the forward orbit of the critical point have been shown todetermine many global dynamical properties of a quadratic polynomial.The article also includes apparently new work on the irreducibilityof iterates of quadratic polynomials.     

Multipliers of Qs spaces and the corona theorem

For s>0, let Q_s be the space of all analytic functions onthe unit disc such that |f'(z)|²(1–|z|²)^s dA(z) isan s-Carleson measure. Here we prove that the corona theoremholds for the algebra of pointwise multipliers of Q_s.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

Geometric Approximation Of The Fibre Of The Freudenthal Suspension

Let Rat_k(CPⁿ) denote the space of based holomorphic maps ofdegree k from the Riemannian sphere S² to the complex projectivespace CPⁿ. The basepoint condition we assume is that f()=[1,..., 1]. Such holomorphic maps are given by rational functions: Rat_k(CPⁿ) ={(p₀(z), ..., p_n(z)):each p_i(z) is a monic, degree-kpolynomial and such that there are no roots common to all p_i(z)}.(1.1) The study of the topology of Rat_k(CPⁿ) originated in [10]. Later,the stable homotopy type of Rat_k(CPⁿ) was described in [3] interms of configuration spaces and Artin's braid groups. LetW(S²ⁿ) denote the homotopy theoretic fibre of the Freudenthalsuspension E:S²ⁿ S²ⁿ⁺¹. Then we have the following sequenceof fibrations: ²S²ⁿ⁺¹ W(S²ⁿ)S²ⁿ S²ⁿ⁺¹. A theorem in [10] tellsus that the inclusion Rat_k(CPⁿ) ²_kCP^{n 2}S²ⁿ⁺¹ is a homotopy equivalenceup to dimension k(2n–1). Thus if we form the direct limitRat(CPⁿ)= lim_k Rat_k(CPⁿ), we have, in particular, that Rat(CPⁿ)is homotopy equivalent to ²S²ⁿ⁺¹. If we take the results of [3] and [10] into account, we naturallyencounter the following problem: how to construct spaces X_k(CPⁿ),which are natural generalizations of Rat_k(CPⁿ), so that X(CPⁿ)approximates W(S²ⁿ). Moreover, we study the stable homotopytype of X_k(CPⁿ). The purpose of this paper is to give an answer to this problem.The results are stated after the following definition. 1991Mathematics Subject Classification 55P35.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

Palindromic matrix polynomials, matrix functions and integral representations

We study the properties of palindromic quadratic matrix polynomials φ(z)=P+Qz+Pz², i.e., quadratic polynomials where the coefficients P and Q are square matrices, and where the constant and the leading coefficients are equal. We show that, for suitable choices of the matrix coefficients P and Q, it is possible to characterize by means of φ(z) well known matrix functions, namely the matrix square root, the matrix polar factor, the matrix sign and the geometric mean of two matrices. Finally we provide some integral representations of these matrix functions.     

The sound field due to a periodic array of forced wave-bearing strips

A compressible fluid in a two-dimensional half-space (y >0) is bounded by a plane surface (y = 0) which is acousticallyhard except for a set of periodically arranged strips S_n givenby nd – a < x < nd + a, y = 0 with n = 0, 1, 2,....The velocity potential Re {(x, y)exp(–it)} satisfies theHelmholtz wave equation in the fluid region y>0, with /y= 0 on the plane y = 0, x S_n. The boundary condition on thepistons S_n is taken to have the form where the prescribed forcing function V(x) is the same on eachstrip, so that V(x + nd) = V(x), and the operators L and M arepolynomial functions of the second derivative ²/x². This boundarycondition includes the possibilities of an elastic plate, amembrane, or an impedance surface for S_n. When the separationdistance d is much greater than the strip width 2a and wavelength2/k, the problem is reduced to that of finding the potential_p due to a single piston S_o set in a rigid baffle, togetherwith a potential _c subject to a similar condition with forcingfunctions exp (ikx) in place of V(x). The problem is generalizedto allow for the possibility of a phased forcing function V(x),such that V(x + nd) = exp (ißnd)V(x), where ßis a given constant.     

A stability criterion

An important technique for determining the stability of a system of ordinary differential equations is to determine whether there are any roots in the positive half-plane of a certain polynomial P(z). Cesari has given a criterion for this in terms of the topological degree of the mapping described by P(z). It is shown here that Cesari's criterion can be reformulated as the problem of approximating the real roots of polynomials which are the real and imaginary parts of the P(z) on certain lines in the z-plane. The roots need only be approxi¬mated closely enough so that their magnitudes can be compared. The derivation of this criterion uses the notion of topological degree but the criterion itself is stated entirely in elementary terms     

On regularity of a quasiconformal mapping at a point of maximal stretching

Local behaviour of a K-quasiconformal mapping f at a point z₀of maximal stretching is studied. A sufficient condition forthe existence of the finite limit lim(f(z) – f(z₀))/(z– z₀)|z – z₀|^1/K–1 as z z₀, and a criterionfor z₀ to be a point of maximal stretching are given.